Chapter 12.2, Problem 46E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Suppose that **a** and **b** are nonzero vectors that are not parallel and **c** is any vector in the plane determined by **a** and **b**. Give a geometric argument to show that c can be written as **c** = *s***a** + *t***b** for suitable scalars *s* and *t*. Then give an argument using components.

To determine

**To give**: A geometric argument to show the vector

Explanation

**Given:**

Non zero and non parallel two-dimensional vectors **a** and **b** and any vector **c**.

**Definition:**

**Scalar multiplication:**

Consider a scalar *c* and a vector **v**. The scalar multiple *c***v**, which is a vector with a length more than **v** in same direction of **v**.

Consider two-dimensional vectors *xy*-plane as they are two-dimensional. Sketch the vectors **a**, **b**, and **c** in *xy*-plane from origin. Name the origin as *O*. Extend the vector **a** using line *A* and vector **b** using *B*.

Draw the parallel lines **a** and **b** from terminal point of vector **c**. Name the intersecting point of lines *A* and *P* and intersecting point of *B* and *Q*.

From explanation, sketch of vectors is shown in Figure 1.

From Figure 1, write the expression for vector **c**.

From Figure 1, write the expression for scalar *s* by the use of definition.

Here,

**a**.

From Figure 1, write the expression for scalar *t* by the use of definition.

Here,

**b**.

From Figure 1, write the expression for vector **c** in terms of vectors **a** and **b.**

The *x* and *y*-coordinates of vector c are estimated by the use of equation (1). Substitution of *x*-coordinated of vectors **a** and **b** along with scalar results *x*-coordinate of vector **c** and substitution of *y*-coordinates of vectors **a** and **b** along with scalar provides the value of *y*-coordinate of vector **c.**

Substitute **a**, **b**, and **c** in equation (1),

Multiply

Substitute **a**, **b**, and **c** in equation (1),

Multiply

Subtract equation (3) from (5)

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P-38RECh-12 P-1PCh-12 P-2PCh-12 P-5P

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